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arXiv:1112.1856 · Physical Review A

Dual quantum-correlation paradigms exhibit opposite statistical-mechanical properties

R. Prabhu, Aditi Sen, Ujjwal Sen

Published Jul 30, 2012• 13 citations
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Abstract

We report opposite statistical-mechanical behaviors of the two major paradigms in which quantum correlation measures are defined, viz., the entanglement-separability paradigm and the information-theoretic one. We show this by considering the ergodic properties of such quantum correlation measures in transverse quantum $XY$ spin-$\frac{1}{2}$ systems in low dimensions. While entanglement measures are ergodic in such models, the quantum correlation measures defined from an information-theoretic perspective can be nonergodic.

DOI
10.1103/physreva.86.012336

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We report opposite statistical-mechanical behaviors of the two major paradigms in which quantum correlation measures are defined, viz., the entanglement-separability paradigm and the information-theoretic one. We show this by considering the ergodic properties of such quantum correlation measures in transverse quantum XYXYXY spin-12\frac{1}{2}21​ systems in low dimensions. While entanglement measures are ergodic in such models, the quantum correlation measures defined from an information-theoretic perspective can be nonergodic.
Quantum correlations are traditionally viewed as constituted out of classical correlations and single-party parameters. While this, of course, remains so, we show that entanglement can have statistical mechanical properties like ergodicity, which are not inherited from the corresponding classical correlations and single-party parameters, in diagonalizable as well as nondiagonalizable low-dimensional quantum spin systems. In particular, we find that the results hold for the quantum XY spin models on the chain, ladder and in two dimensions.
We investigate the time dynamics of quantum correlations of the anisotropic Heisenberg model in a time-dependent magnetic field in one-dimensional, ladder, and two-dimensional lattices. We find that quantum correlation measures in the entanglement-separability paradigm are ergodic in these systems irrespective of system parameters. However, information-theoretic quantum correlation measures can also be nonergodic and exhibit a transition from nonergodic to ergodic behavior with the change of interaction strength in the direction of the magnetic field. We also observe that the transition point changes drastically as we go from one-dimensional and ladder lattices to the two-dimensional one.
There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.
We investigate the static and dynamical patterns of entanglement in an anisotropic XYXYXY model with an alternating transverse magnetic field, which is equivalent to a two-component one-dimensional Fermi gas on a lattice, a system realizable with current technology. Apart from the antiferromagnetic and paramagnetic phases, the model possesses a dimer phase which is not present in the transverse XYXYXY model. At zero temperature, we find that the first derivative of bipartite entanglement can detect all the three phases. We analytically show that the model has a ``factorization line'' on the plane of system parameters, in which the zero-temperature state is separable. Along with investigating the effect of temperature on entanglement in a phase plane, we also report a nonmonotonic behavior of entanglement with respect to temperature in the antiferromagnetic and paramagnetic phases, which is surprisingly absent in the dimer phase. Since the time dynamics of entanglement in a realizable physical system plays an important role in quantum information processing tasks, the evolutions of entanglement at small as well as large time are examined. Consideration of large-time behavior of entanglement helps us to prove that in this model, entanglement is always ergodic. We observe that other quantum correlation measures can qualitatively show similar features in zero and finite temperatures. However, unlike nearest-neighbor entanglement, the nearest-neighbor information-theoretic measures can be both ergodic as well as nonergodic, depending on the system parameters.
Typically in many-body systems the correlation energy, which is defined as the difference between the exact ground-state energy and the mean-field solution, has been a measure of the system's total correlations. However, under the quantum information context, it is possible to define some quantities in terms of the system's constituents that measure the classical and quantum correlations, such as the entanglement entropy, mutual information, quantum discord, one-body entropy, etc. In this work we apply concepts of quantum information in fermionic systems in order to study traditional correlation measures (the relative correlation energy). Concretely, we analyze the two- and three-level Lipkin models, which are exactly solvable (but nontrivial) models commonly used in the context of the many-body problem. We conclude that the correlation energy is not a good estimator of the total correlation of the system, and information theoretic measures should be considered.
We study the ground state quantum correlation of Ising model in a transverse field (ITF) by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomena and scaling behavior when the size of the systems becomes large. Our results also indicate a universal behavior of the critical exponent of ITF under QRG theory that the critical exponent of different measures is identical, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion including the entanglement-separability paradigm and the information-theoretic paradigm have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.